Wave_Functions_and_Probability

# Wave_Functions_and_Probability - WAVE FUNCTIONS AND...

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Unformatted text preview: WAVE FUNCTIONS AND PROBABILITY 1 I: Thinking about the wave function In quantum mechanics, the term wave function usually refers to a solution to the Schr¨ odinger equation, i ~ ∂ Ψ( x,t ) ∂t =- ~ 2 2 m ∂ 2 Ψ( x,t ) ∂x 2 + V ( x )Ψ( x,t ) , where V ( x ) is the potential energy experienced by a particle of mass m and Ψ( x,t ) is the wave function in this one-dimensional example. A. Let’s say you have a system where the wave function is of the form: Ψ 1 ( x,t ) = f ( x ) e iωt where f ( x ) is some real-valued function of x . 1. Is | Ψ 1 ( x,t ) | 2 real? Is it positive? Do your answers make sense given the physical meaning (as discussed in class) of | Ψ 1 ( x,t ) | 2 ? 2. Does Ψ 1 ( x,t ) depend on time? Does | Ψ 1 ( x,t ) | 2 depend on time? 3. Write down an expression for h x i . Does it depend on time? Is it real? Describe in words how you interpret this quantity. Precisely what information do you get from h x i ? 4. Write down an expression for h g ( x ) i where g ( x ) is any real-valued function of x . Does it depend on time? Again, how would you physically interpret h g ( x ) i (hint: think about what you would actually measure)? PHYS 3220 Tutorials 2008 - 2010 c S. Goldhaber, S. Pollock, and the Physics Education Group University of Colorado, Boulder WAVE FUNCTIONS AND PROBABILITY 2 B. Now let’s say your system is a bit more complex (pun intended): Ψ 2 ( x,t ) = f ( x ) e iωt + g ( x ) e 2 iωt...
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Wave_Functions_and_Probability - WAVE FUNCTIONS AND...

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